Bayesian Estimation for Spherically Symmetric Distributions

The theory of Bayesian least squares is developed for a general and more tangible notion of conjugacy than in models which make the more conventional assumption of normality. This paper is primarily concerned with extending the results of classical conjugate normal-normal Bayesian analysis to the canonical setting of the generalized linear model when, at the same time, the sampling distribution and the prior are spherically symmetric. In order to underline the intrinsic aspect of our results, the approach of multivariate analysis adopted here is coordinate free. Examples which illustrate the theory are also presented.